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Trigonometry and Music


Date: 12/21/98 at 09:57:17
From: Elizabeth
Subject: Trigonometry and music

How is trigonometry used in music?


Date: 12/21/98 at 13:04:53
From: Doctor Santu
Subject: Re: Trigonometry and music

Dear Elizabeth,

Certainly the functions sine and cosine have a connection to music.

You probably know that sound consists of waves, and that sin(x) and 
cos(x) are waves. Fourier, a French heat engineer, wondered whether it 
might be possible to write *any* wave in terms of combinations of sines 
and cosines; and it turned out that, yes, it's possible (provided your 
wave is smooth and well-behaved enough).

So, for instance, a wave might be representable as a combination such 
as  1.4 cos x + 2.8 sin x - 1.103 cos 2x + 3.32 cos 3x + 5.6 sin 3x.

Normally, a wave selected at random can only be represented by an 
infinite number of such terms; in other words, the combination doesn't 
stop, but goes on forever, though the numbers in front of the sines 
and cosines get smaller and smaller.

Musically, when an instrument plays a note, the basic note can be 
represented by A.cos Lt + B.sin Lt. The number L has to do with how 
high the note is (pitch), and A and B have to do with how loud it is; 
t is time.

But that function is a "pure tone" that sort of sounds like a flute. 
The unique sounds of oboes and clarinets and so forth are due to almost 
inaudible notes at the octave, octave+fifth, etc. 

The Dr. Math archives contain a reply sent to a similar question 
earlier this year:

http://mathforum.org/dr.math/problems/angelica12.3.97.html   

Hope it's useful!

- Doctor Santu, The Math Forum
  http://mathforum.org/dr.math/   


Date: 06/30/2002 at 21:54:29
From: Dan
Subject: Trigonometry and Music

I am aware that sound waves can be described through trigonometric 
functions, in equation form. I am interested in how their geometric 
shapes relate to their trigonometric functions. Why are certain 
combinations of tones more pleasing to the ear than others? Can the 
trigonometric function for a sound wave be used to describe light, 
radio, X-ray waves?  How can we derive the function for a wave? 
(example--sound wave)

Brian


Date: 07/16/2002 at 09:10:44
From: Doctor Nitrogen
Subject: Re: Trigonometry and Music

Hello, Brian:

If you look at the graph for the sine and cosine functions, you will 
see that they have all the characteristics of a wave traveling along 
the x-axis. Just like a wave, they have amplitude, and they are 
periodic, and even have a frequency.

There is a branch of Mathematics called Fourier Analysis/Fourier 
Series, which uses this property of the sine and cosine graphs, to 
decribe wave motion. Frequently Fourier Series are used to analyze the 
wave properties of a vibrating string, like a violin string, or the 
vibrational modes of a drum membrane. When Fourier Series are used 
this way, they are usually solutions to Partial Differential 
Equations.

Sound waves are longitudinal, rather than transverse, like water 
waves, or waves along a string fixed at both ends and set to vibrate, 
but sine and cosine waves describe sound waves also.

The reason certain tones are more pleasing to the ear than others is 
that when partial differential equations describing wave motion are 
solved by Fourier Series, the solutions have a special number 
associated with them for the modes of vibration for the system. When 
the vibrational modes are integral multiples of a fundamental mode of 
vibration in the solution, you get music. When they are not, you 
don't. That's why you can get harmonics on a violin, flute, a pipe 
organ, or a piano, but not on a snare or bass drum. So classical 
violinist Itzhak Perlman might be creating music with his violin, but 
the rock star Bon Jovi's drummer is not.

Light waves were first described by a physicist named James Clerk 
Maxwell. When he solved one of his partial differential equations he 
got a wave motion for light. A branch of physics called Quantum 
Mechanics, however, was developed around 1900 to explain certain 
things about light waves that Maxwell's equations could not. Can you 
believe it was once thought that because light was a wave there should 
be no limit to the frequency of the wave in a physical system? This 
meant that for ordinary light trapped in a box lined with mirrors, the 
light could eventually escape as gamma rays! This paradox was called 
the Ultraviolet Catastrophe. So to answer your question about light, 
Quantum Mechanics has replaced the old ideas about the wave nature of 
light and replaced them with the idea that sometimes light behaves 
like a particle, and sometimes like a wave.

In Quantum Mechanics, applied to the hydrogen atom, when an electron 
"jumps" from a higher energy level to a lower one, the light energy it 
radiates as a "photon" particle is:

   E_2 - E_1 = n*hbar*nu, n = 1, 2, 3, ....

where hbar is a constant called h, divided by 2*pi, and nu is the 
frequency of the photon.

Even in Quantum Mechanics there is a wave equation, called the 
Schroedinger Wave Equation.

I hope this has answered your questions.

- Doctor Nitrogen, The Math Forum
  
    
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